We develop a duopoly model in which firms compete for the market (e.g., investing in process innovation or product development) as well as in the market (e.g., setting quantities or prices). Competition for the market generates multiple equilibria that differ in the firms’ investment levels, relative size, and profitability. We show that monopolization that affects competition in the market can act as an equilibrium selection device in competition for the market. In particular, it eliminates equilibria that are undesirable for the monopolizing firm, while not generating new equilibria. This result complicates the task of determining whether a firm's dominance in a given market is the result of fair competition or unlawful monopolization. We discuss a number of implications for antitrust policy and litigation, and illustrate these by means of two well-known antitrust cases.